EQUILIBRIUM OF A PARTICLE 27 PROBLEMS. 1. Three men pull on a ring. The first man pulls with a force of 50 pounds toward the N. 30° W. The second man pulls toward the S. 45° E. with a force of 75 pounds, and the third man pulls with a force of 100 pounds toward the west. Determine the magnitude and direction of the resultant force. 2. Show that the resultant of two forces acting upon a particle lies in the plane of the given forces. 3. Show that the line of action of the resultant of two forces lies within the angle made by the forces. 4. Find the direction and magnitude of the resultant of three equal forces which act along the axes of a rectangular system of coordinates. " GENERAL PROBLEMS. 1. A particle is in equilibrium under the action of the forces P, Q, and R. Prove that _, sin (Q, R) sin (P, R) sin (P, Q) ' where (Q, R), etc., denote the angles between Q and R, etc. 2. Two particles of weights W\ and TFo rest upon a smooth sphere of radius a. The particles are attached to the ends of a string of length lt which passes over a smooth peg vertically above the center of the sphere. If h is the distance between the peg and the center of the sphere, find (1) the position of equilibrium of the particles, (2) the tensile force in the string, and (3) the reaction of the sphere. 3. The lengths of the mast and the boom of a derrick are a and 6 respectively. Supposing the lunges at the lower end of the boom and the pulley at the upper end to be smooth, find the angle the boom makes with the vertical when a weight W is suspended in equilibrium. 4. Find the tensile force in the chain and the compression in the boom of the preceding problem. 6. Two rings of weights Wi and W^ are held on a smooth circular wire in a vertical plane by means of a string subtending an angle 2 a. at the center. Show that the inclination of the string to the horizon is given by - W, 6. A bridge, Fig. (a) , of 60-foot span and 40-foot width has two queen-post trusses 9 feet deep. Each truss is divided into three equal parts by two